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# 11.5.2: Prandtl's Condition

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It can be easily observed that the temperature from both sides of the shock wave is discontinuous. Therefore, the speed of sound is different in these adjoining mediums. It is therefore convenient to define the star Mach number that will be independent of the specific Mach number (independent of the temperature).

$M^{*} = \dfrac{ U }{ c^{*} } = \dfrac{c }{ c^{*} } \dfrac{U }{ c} = \dfrac{c }{ c^{*} }\, M \label{shock:eq:starMtoM}$
The jump condition across the shock must satisfy the constant energy.
$\dfrac{c^2 }{ k-1} + \dfrac{U^2 }{ 2 } = \dfrac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Civil_Engineering/Book:_Fluid_Mechanics_(Bar-Meir)/11:_Compressible_Flow_One_Dimensional/11.5_Normal_Shock/11.5.2:_Prandtl's_Condition), /content/body/p[2]/span, line 1, column 2 ^2 \label{shock:eq:momentumC}$
Dividing the mass equation by the momentum equation and combining it with the perfect gas model yields
${{c_1}^2 \over k\, U_1 } + U_1 = {{c_2}^2 \over k\, U_2 } + U_2 \label{shock:eq:massMomOFS}$

Combining equation (29) and (30) results in

$\dfrac{1 }{ k\,U_1} \left[ \dfrac{k+1 }{ 2 }\, {c^{*}}^2 - {k-1 \over 2 } U_1 \right] + U_1 = \dfrac{1 }{ k\,U_2} \left[ {k+1 \over 2 } {c^{*}}^2 - \dfrac{k-1 }{ 2 }\, U_2 \right] + U_2 \label{shock:eq:combAllR}$

After rearranging and dividing equation (31) the following can be obtained:

$U_1\,U_2 = {c^{*}}^2 \label{shock:eq:unPr}$ or in a dimensionless form

${M^{*}}_1\, {M^{*}}_2 = {c^{*}}^2 \label{shock:eq:PrDless}$

## Contributors and Attributions

• Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.

This page titled 11.5.2: Prandtl's Condition is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

This page titled 11.5.2: Prandtl's Condition is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Genick Bar-Meir via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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